# Complex Analysis: Set Theory

I just had a confusing lecture on the following:

Open Set

Closed set

Open and Connected

Domain

Bounded and Unbounded sets

The lecture was confusing to say the least. Could you please give me a clear explanation of these concepts with examples?

By the Way:

Heisenberg and Schrodinger are in a car and are stopped by police officers. One officer asks Heisenberg if he knows how fast he is going. He answers, "No but I know where I am."

The other asks Schrodinger if he is aware of the dead cat in his trunk. He answers, " I am now."

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#### Solution Preview

The set S in the complex plane define some Euclidean area.

We first define the concept of neighborhood. A neighborhood of a point z is the set that contains all points such that

z is an interior point of the set if there exists such that

An interior point of S is a point which is included in S and all the points around it within an infinitesimally small radius are also members of the set S.

All the interior points of S form a set called the interior of S and is denoted by

Now, if we have a set then we have a complementary set , that is if then and vice versa. If a point belongs to S it cannot belong to SC, and if a point belongs to SC it cannot belong to S.

An exterior point of S is defined as an interior point of SC.

That is:

iff

in other words, is an exterior point of S if an only if - there are no points in the neighborhood of z that belong to S.

A boundary point of S is a point which is not interior nor exterior to S, that is, in its neighborhood we find points that belong to S and points that belong to

The set of all such points is called the ...

#### Solution Summary

The expert examines the complex analysis set theory.